Follow-me robotics has gotten genuinely affordable in the last 18 months — UWB (Ultra-Wide Band) modules dropped to under $15 each and provide centimeter-accurate ranging at 1Hz indoors. That's enough to build a robot that follows you at walking pace without a camera, without GPS, and without complex SLAM. The minimum-viable version is two motorized wheels, two casters, and a UWB anchor on a small servo gimbal that points where the tag in your pocket is.
The cargo case for grocery carts is real: lots of people have a long walk from the car to the kitchen door, multiple trips, and a manual pull-cart that wobbles. A 30 kg payload with a 50W per wheel hub-motor base handles loaded shopping bags up a 10° driveway slope. Speed is intentionally walk-paced (1-1.4 m/s) — faster and it becomes scary near children and dogs, slower and it lags behind you.
The non-obvious safety design is the front bumper. Even with UWB tracking, the cart can run into your heel if you stop suddenly. A simple horizontal IR proximity sensor on the front (Sharp GP2Y0A21 or VL53L0X ToF) brakes the motors hard if anything is within 35cm — your legs, a curb, the dog. Layered on top of that, a hardware kill-switch on the handle means you can grab the cart and stop it manually with zero firmware involvement.